Mathematical Tripos II Numerical Analysis (Michaelmas 2021)

Lecturer: Hamza Fawzi

Tue-Thu-Sat 10am - MR4

Lecture notes

Lecture 1: Poisson's equation, finite differences, and the five-point formula
Lecture 2: Five-point formula (continued)
Lecture 3: Convergence
Lecture 4: FFT for Fast Poisson solver + Diffusion equation
Lecture 5: Stability
Lecture 6: Crank-Nicolson
Lecture 7: Stability using Fourier analysis
Lecture 8: The advection equation
Lecture 9: The diffusion equation in two space dimensions
Lecture 10: Splitting schemes
Lecture 11: Spectral methods (1) - Fourier approximation
Lecture 12: Spectral methods (2)
Lecture 13: Spectral methods (3) - Chebyshev approximation
Lecture 14: Spectral methods (4) - Equations of evolution
Lecture 15: Iterative methods for linear systems - Jacobi, Gauss-Seidel
Lecture 16: Iterative methods for linear systems - convergence, relaxation
Lecture 17: Iterative methods for linear systems - multigrid methods
Lecture 18: Iterative methods for linear systems - steepest descent
Lecture 19: Iterative methods for linear systems - conjugate gradient
Lecture 20: Iterative methods for linear systems - CG convergence and preconditioning
Lecture 21: Eigenvalues and eigenvectors - Power iteration, inverse iteration, Rayleigh quotient iteration
Lecture 22: Eigenvalues and eigenvectors - Simultaneous and QR iteration
Lecture 23: Eigenvalues and eigenvectors - QR iteration: shifts, and deflation
Lecture 24: Eigenvalues and eigenvectors - Reduction to tridiagonal form, nonsymmetric matrices

The notes and slides are heavily inspired by last year's edition of the course, taught by Prof. Anders Hansen (http://www.damtp.cam.ac.uk/research/afha/lectures/Part_II_NumAn/).

Example sheets

Other resources

IB Numerical Analysis course
Matlab demos
A first course in the numerical analysis of differential equations, A. Iserles, Cambridge University Press [The course mostly follows this book]
Iterative methods for sparse linear systems, Y. Saad
A multigrid tutorial, W. L. Briggs, V. E. Henson, S. F. McCormick
The definition of numerical analysis by L. N. Trefethen
Top ten algorithms of the 20th century, by Barry Cipra